Expand the Logarithmic Expression log of ((5 square root of u)/(w^2))^3

$\log ({(\frac{5 \sqrt{u}}{w^{2}})}^{3})$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{u}$ as $u^{\frac{1}{2}}$ .

$\log ({(\frac{5 u^{\frac{1}{2}}}{w^{2}})}^{3})$

Expand $\log ({(\frac{5 u^{\frac{1}{2}}}{w^{2}})}^{3})$ by moving $3$ outside the logarithm.

$3 \log (\frac{5 u^{\frac{1}{2}}}{w^{2}})$

Rewrite $\log (\frac{5 u^{\frac{1}{2}}}{w^{2}})$ as $\log (5 u^{\frac{1}{2}}) - \log (w^{2})$ .

$3 (\log (5 u^{\frac{1}{2}}) - \log (w^{2}))$

Expand $\log (w^{2})$ by moving $2$ outside the logarithm.

$3 (\log (5 u^{\frac{1}{2}}) - (2 \log (w)))$

Multiply $2$ by $- 1$ .

$3 (\log (5 u^{\frac{1}{2}}) - 2 \log (w))$

Rewrite $\log (5 u^{\frac{1}{2}})$ as $\log (5) + \log (u^{\frac{1}{2}})$ .

$3 (\log (5) + \log (u^{\frac{1}{2}}) - 2 \log (w))$

Expand $\log (u^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$3 (\log (5) + \frac{1}{2} \log (u) - 2 \log (w))$

Combine $\frac{1}{2}$ and $\log (u)$ .

$3 (\log (5) + \frac{\log (u)}{2} - 2 \log (w))$

Apply the distributive property.

$3 \log (5) + 3 \frac{\log (u)}{2} + 3 (- 2 \log (w))$

Simplify.

Combine $3$ and $\frac{\log (u)}{2}$ .

$3 \log (5) + \frac{3 \log (u)}{2} + 3 (- 2 \log (w))$

Multiply $- 2$ by $3$ .

$3 \log (5) + \frac{3 \log (u)}{2} - 6 \log (w)$